On Equitable Resolving Domination in Some Path Related Graphs

For a graph G with vertex set V(G), an ordered subset W={w_1,w_2,...,w_k} of V(G), the k-vector (d(v,w_1),d(v,w_2),...,d(v,w_k)) is called the resolving vector of v∈V(G) with respect to W denoted by r(v|W). The set W is called a resolving set of graph G if r(u|W)≠r(v|W) for any two distinct vertices u and v. A subset D of V(G) is called a resolving dominating set if it is resolving as well as dominating and it is called equitable if for every vertex v∈V(G)-D, there exists a vertex u∈D such that uv∈E(G) and |deg(u)-deg(v)|≤1. A dominating set which is both equitable and resolving is called equitable resolving dominating set. The minimum cardinality of an equitable resolving dominating set is called an equitable resolving domination number of G which is denoted by γ_rs^e (G). In this paper, we determine exact values of equitable resolving domination number of some path related graphs.